文章目录
几何定义 Vs. 代数定义
前些時候我們講一個幾何定義叫metrical of scalar product and vector product. From this we proved the distributive law. Then we had algebraic definition.（前段时间我们讲了一个几何定义，叫做标量积和向量积的度量。由此我们证明了分配律。然后我们有了代数定义。）
用這個邏輯就不是一個definition了, 這是一個formula. 這個的好處是from the beginning, the geometrical definition does not depend on the coordinate choice. 兩個vector的scalar product 定義裡頭隨便是什麼coordinate choice沒關係。 所以這個algebraic definition, algebraic formula 跟剛才這樣子是與choice of coordinate沒關係。 Of course you can ask can I start from the algebraic formula as the definition? In fact, in many books, the scalar product and the vector product are defined by the algebraic formula.（按照这个逻辑，它就不再是一个定义了，而是一个公式。这样做的好处是从一开始，几何定义就不依赖于坐标系的选择。在两个向量的标量积定义中，无论选择什么坐标系都无关紧要。因此，这个代数定义、代数公式与刚才提到的坐标系选择无关。当然，你可能会问，我能否以代数公式作为定义的起点？事实上，在许多书籍中，标量积和向量积都是通过代数公式来定义的。）
But if you do it that way, you need to prove that according to that definition, you get a scalar product, which however you have to prove is independent of the choice of coordinates. 那個proof比較麻煩。 那個proof基本上就是說, if you rotate the axis, it will not change the scalar product. To prove from the algebraic definition that the scalar product is independent of the choice of coordinates.（但如果你那样做，就需要证明根据该定义，你得到的是一个标量积，而且必须证明这个标量积与坐标的选择无关。那个证明比较麻烦。那个证明基本上就是说，如果你旋转坐标轴，标量积不会改变。要从代数定义证明标量积与坐标的选择无关。）
几何定义 → 代数公式：从几何意义定义标量积和向量积，得到代数公式作为推论，而非定义天然地与坐标系选择无关；
代数定义 → 几何意义：直接用坐标分量的代数公式作为定义，必须额外证明结果与坐标系选择无关，证明难点是需要证明坐标轴旋转不改变标量积的值
To prove from the algebraic definition that the scalar product is independent of the choice of coordinates needs a discussion of symmetry. 因為你從一個coordinate system換成對過來的coordinate system, 這兩個之間的關係是一個叫做symmetry consideration。 那麼這個, once you know how to do it, it's actually also very simple. But that's not something that we are going to explore.（要从代数定义证明标量积与坐标系的选择无关，需要讨论对称性。因为当你从一个坐标系转换到另一个坐标系时，这两者之间的关系涉及所谓的对称性考量。一旦你掌握了方法，实际上也非常简单。但这并不是我们要探讨的内容。）
But I want to mention that this symmetry concept, 对称, has become one of the central subjects in physics. And it's likely to remain so in the 21st century. We'll have the occasion to come back to this subject of symmetry in the future.（但我想指出的是，这种对称性概念已成为物理学的核心课题之一。这一概念很可能在21世纪仍占据重要地位。未来我们还将有机会重新探讨对称性这一主题。）
有效数字
有同学问我关于这个significant figures。 If you have a product or division, 比如說, 假如我寫3.24乘上, 點零二三六八九, divided by 一百四十六, the important thing is to realize that this 八九 is not useful. Because the 3.24 is only accurate up to the last digit, the 4. And a small change, a one-unit change in 4, would affect this one. So, for a product, try to keep the number of significant figures the same.（有同学问我关于有效数字的问题。比如在乘法或除法运算中，假设我们计算 3.24 乘以 0.023689，再除以 146。关键是要意识到最后的 "89" 没有实际意义。因为 3.24 的精度只精确到最后一位数字 "4"。如果 "4" 发生一个单位的变化，就会影响整个结果。因此，进行乘除运算时，最终结果应保留与原始数据中有效数字位数最少的那个数相同的有效位数。）
In other words, try to make every one three significant figures. That means, 你不必寫這兩個。 Does everybody understand this? 為什麼要那麼多? 假如你有一個數目,一共只有兩位, 那你剩下的那些五位沒用處的。 因為那兩位就代表說是已經只有百分之一的accuracy。 你到後面那個特別accurate, 是沒有用處的。 所以, for a product, try to keep the number of significant figures the same. In this case, three.（换句话说，尽量让每个数字保留三位有效数字。这意味着，你不必写这两个。大家都明白了吗？为什么要那么多？假如你有一个数字，总共只有两位数，那你后面那些五位数字是没有用的。因为那两位数就表示只有百分之一的精确度。你后面那些特别精确的数字是没有用的。所以，对于一个产品，尽量保持有效数字的位数相同。在这里，就是三位。）
But for addition, it's an entirely different story. For example, if I write 3.24 plus 0,2,3,4,5, 在這裡請... OK, let's do it this way. This one has five figures.（但对于加法来说，情况就完全不同了。例如，如果我写 133.24+0.02345，在这里请...好吧，我们这样做。这个有五个数字。）
就是加上。 133.24有五個 figure。 這裡也寫五個figure。 But actually, 到了這個.24以後, 到了這個.2以後, 這些figure都沒用處。 For the same reason, because you may change this by one, and that would affect this one, not this. So therefore, in this case, 這些都不需要。 You can write it there, but it has no meaning. So, in this case, you don't count the number of significant figures. You ask what is the largest inaccuracy, and that determines how many figures each one should be kept. 大家都懂了這個沒有? (比如这个 133.24 有五个有效数字。但事实上，第二项到了小数点后，尤其是 .2 之后------这些数字（指 0.02345 中 "345"）都没用。原因相同：因为哪怕你把这个数字最后改一个单位（比如 ±1），也只影响它自己这部分，而不影响整体。所以这种情况下，这些多余的数字不需要保留。你可以写在那里，但它们没有实际意义。因此，这种情况下不需要计算有效数字位数。你问最大的误差是多少，这决定了每个数值应该保留多少位有效数字。大家都明白了吗？)
有效数字的运算规则：
- 加法和减法：结果的小数点后位数应该与参与运算的数中小数点后位数最少的数相同
- 乘法和除法：结果的有效数字位数应该与参与运算的数中有效数字位数最少的数相同
有效数字的处理遵循 木桶效应 ------结果的精度由最不精确的数据决定。一些示例如下：
python
12.34 (小数点后2位)
+ 1.2 (小数点后1位) ← 最少
---------
13.54 → 13.5 (保留1位小数)
123.456 (小数点后3位)
+ 0.12 (小数点后2位) ← 最少
- 10.1 (小数点后1位) ← 最少
-----------
113.476 → 113.5 (保留1位小数)
2.34 × 1.2 = 2.808
↓ ↓
3位 2位 ← 最少
结果：2.8 (保留2位有效数字)
123.45 ÷ 2.1 = 58.785...
↓ ↓
5位 2位 ← 最少
结果：59 (保留2位有效数字)
(12.3 + 1.27) × 2.1
第一步：加法 12.3 + 1.27 = 13.57 → 13.6 (保留1位小数)
第二步：乘法 13.6 × 2.1 = 28.56 → 29 (保留 2 位有效数字)练习题
我們現在來... We'll do an exercise. 書上有這麼一個exercise。 這是很簡單的exercise。 有一個uniform circular motion。 它的r是等於5.0個meter。 它的acceleration是7個g。 When you train a spaceman, a man who is going to fly in the spaceship, one of the important things is to make him realize that when he is accelerated and when he is decelerated, he suffers a tremendous force.(我们现在来... 我们来做一道练习题。书上就有这么一道题。这是一道很简单的题。有一个匀速圆周运动。它的半径等于 5.0 米。它的加速度是 7 个 g。当你训练宇航员，也就是即将乘坐宇宙飞船飞行的人时，其中一个重要事项就是让他意识到，当他加速或减速时，会承受巨大的力。)
因为要把火箭送上太空，需要用很大的推力。这个推力会产生很大的加速度。平时我们只能感受到 1 1 1 个 g g g 的加速度，但火箭发射时需要承受多个 g g g。所以必须进行训练。怎么训练呢？用一个很大的转盘，把人固定在里面...可以说是嵌在转盘内部。然后快速旋转。旋转时会产生向心力。就像我们上节课讲的，这个向心力会让人...如果人在转盘边缘...转盘会给人一个向内的力，产生加速度。这个力可以达到很大的 g g g 值。假设是 7 g 7g 7g，我们来计算此时的速度 v v v，这就是题目要求。我们知道加速度 a = v 2 / r a=v^2/r a=v2/r。所以速度 v = r a v=\sqrt{ ra} v=ra ，即 5.0 m×7×g 开方。g 取 9.78 m/s 2 \text{m/s}^2 m/s2。先看量纲对不对？ m × ( m / s 2 ) m×(m/s^2) m×(m/s2) 得到 m 2 / s 2 m^2/s^2 m2/s2，开根号后是 m / s m/s m/s，量纲正确。具体数值计算这里就不展开了。
中子星
好，我们再做第四章第 50 题。根据牛顿定律：普通物质由原子组成，原子包含原子核和围绕核运动的电子。
If you try to squeeze the electrons toward the center, this is one of the great achievements of 20th century physics. One understood why it is difficult to squeeze an atom to make the orbits smaller. However, if you have tremendous forces, you would eventually be able to squeeze it. And eventually, you can squeeze all the electrons into the nucleus. But the nucleus consists of protons and neutrons. If you squeeze an electron into it, it becomes protons, neutrons and electrons.(如果你试图将电子挤压到中心，这是 20 世纪物理学的一项伟大成就。人们明白了为什么难以通过挤压原子来使轨道变小。然而，如果你有巨大的力量，最终你就能将其挤压。最终，你可以把所有电子都挤进原子核里。但原子核由质子和中子组成。如果把一个电子挤进去，它就会变成质子、中子和电子。)
Then the electron and the proton would combine to form neutrons. So eventually, the whole atom would become a very small thing. And that small thing consists of only neutrons. And eventually, these neutrons emerge together. It becomes a neutron liquid. And that's called a neutron star.(然后电子和质子会结合形成中子。最终，整个原子会变成一个非常小的东西。而这个微小的东西仅由中子组成。最终，这些中子聚集在一起。它变成了中子流体。这就是所谓的中子星。)
The neutron star is very dense. 密度非常之大。 A typical neutron star has a radius of 20 kilometers. You can imagine that it is very small. The current land has a diameter of 8,000 miles. So this neutron star is a very small thing. If such a small thing has a large mass, its density will be very large.(中子星的密度非常高。一颗典型中子星的半径仅有 20 公里。你可以想象它有多小。目前地球的直径是 8000 英里。所以这颗中子星是个非常小的天体。如果这么小的天体拥有巨大质量，其密度自然就会极其惊人。)
This neutron star usually rotates. Its period is one second. It rotates very fast. This is a very strange thing. Its density is very large, very small, very heavy, and it rotates very fast. We can calculate the relationship between the period and omega.(这颗中子星通常会旋转。它的周期为一秒，转速非常快。这是一件非常奇怪的事情。它的密度非常大，体积非常小，质量非常重，而且旋转速度极快。我们可以计算出周期与角速度之间的关系。)
T = 2 π ω T=\frac{2\pi}{\omega} T=ω2π
Omega is the angular momentum, which is the angular velocity. Omega times t means omega is the radius traveled in one second. That much of the time is the radius traveled in one revolution. One revolution is 2 pi. So I have this formula. It is in one second.( ω \omega ω 是角动量，也就是角速度，Omega 乘以 t 表示， ω \omega ω 是一秒钟内移动的半径。那么多时间就是一转内移动的半径。一转是 2π。所以我得到了这个公式。它是在一秒钟内完成的。)
Therefore, if we calculate the velocity, the velocity is R omega times R times 2 pi over t. So we can calculate this. If we add this up, we get 12 meters per second. We can also calculate the acceleration.（因此，如果我们计算速度，速度等于R乘以ω再乘以R乘以2π除以t。这样我们就能算出结果。把这些数值相加，我们得到每秒 12 米。我们还可以计算加速度。）
v = ω r = r 2 π T v=\omega r=\frac{r2\pi}{T} v=ωr=Tr2π
Acceleration is equal to V square over R. This calculation is equal to 72. So these are very simple exercises. （加速度等于 V平方除以 R。这个计算结果等于 72。这些都是非常简单的练习。）
第四章之相对运动（4.6、4.7）
Next, I'll talk about relative motion.This is a coordinate system A, there is a coordinate system B, then there is a point P, then R, P, A, this notation, you read this notation, the position of R, the position of the point P in the coordinate A, the first one indicates the thing you are talking about, in this case P, the second tells you about the reference, so it is relative to this system. Then this is this, this is this plus this, so it is equal to R of B relative to A plus R of P relative to B. It is better if I write it this way. You can also write it this way.（接下来，我将讨论相对运动。这是一个坐标系 A，还有一个坐标系 B，然后有一个点 P，然后是R、P、A，这个符号，你读这个符号，R 的位置，点 P 在坐标系 A 中的位置，第一个表示你正在谈论的东西，在这里是 P，第二个告诉你参考系，所以它是相对于这个系统的。然后这是这个，这是这个加上这个，所以它等于 B 相对于 A 的 R 加上 P 相对于 B 的 R。我这样写更好。你也可以这样写。）
r ⃗ P A = r ⃗ B A + r ⃗ P B \vec{r}{PA} = \vec{r}{BA} + \vec{r}_{PB} r PA=r BA+r PB
The important thing is this matches this, this matches this, this matches this, and start with P and end with A. Is that clear? Yes. OK. What I just said is very simple, so now let's do something more complicated.（重要的是这个匹配这个，这个匹配这个，这个匹配这个，以 P 开头，以 A 结尾。明白了吗？是的。好的。我刚才说的很简单，现在让我们做一些更复杂的事情。）
General planar motion. Planar means planar, P-L-A-N-A-R is a derivative of plane, P-L-A-N-E. Planar is plane, planar is plane. There is a big difference between Chinese and Western words. Chinese wants to be concise, so many of the adjectives in Chinese are cancelled. I just said it is plane, but when speaking, they cancel it all.(一般平面运动。平面就是平面，P-L-A-N-A-R是plane（平面）的派生词，P-L-A-N-E。平面就是平面，平面就是平面。中西方的语言有很大差异。中文追求简洁，所以中文里的很多形容词都被省略了。我刚才说它是平面，但在说话时，它们都被省略了。)
平面运动（不一定匀速，不一定圆周或直线运动）
!warning
⚠️：劝退警告，下面内容开始上强度了，
So it becomes plane motion. In fact, planar motion means planar motion, a normal planar motion. What does that mean? We studied it last time. If it is a uniform circular motion, we have discussed that. Now we are going to discuss a general motion. It is not necessarily uniform, and it is not necessarily circular.(所以这就变成了平面运动。实际上，平面运动指的是平面运动，一种正常的平面运动。这是什么意思呢？我们上次已经研究过了。如果是匀速圆周运动，我们已经讨论过了。现在我们要讨论的是一般的运动。它不一定是匀速的，也不一定是圆周的。)
But it is in a plane. So there is a point here, which is P. P moves in this direction. It has an R and a C. We have to write its coordinate. I use polar coordinate r theta to describe the motion. So I write it as r cosine theta, r sine theta. I could have written it as r cosine theta i plus r sine theta j. That is what we did last time.(但这是在平面上。所以这里有一个点，就是P。P沿着这个方向移动。它有一个R和一个 θ \theta θ。我们需要写出它的坐标。我用极坐标 r r r 和 θ θ θ 来描述这个运动。所以我写成 ( r cos θ , r sin θ ) (r\cosθ, r\sinθ) (rcosθ,rsinθ)。我也可以写成 r cos θ i ^ + r sin θ j ^ r\cos θ \hat{i} +r \sin θ \hat{j} rcosθi^+rsinθj^。这就是我们上次所做的。)
r ⃗ = ( r cos θ , r sin θ ) \vec{r}=(r\cos \theta, r \sin \theta) r =(rcosθ,rsinθ)
But for convenience, I change the notation to x coordinate here, y coordinate here. Now I am going to calculate its derivative, its velocity. Velocity is the derivative of r with respect to t. So we take r cosine theta as the derivative.(但为了方便起见，我将这里的符号改为x坐标，这里是y坐标。现在我要计算它的导数，也就是速度。速度是r对t的导数。所以我们取r乘以cosθ作为导数。)
在理论物理中，对时间求导通常在相应量的头顶上加个 点，例如：
v ⃗ = d r ⃗ d t = r ⃗ ˙ \vec{v}=\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}=\dot{\vec{r}} v =dtdr =r ˙
但需要注意，只有对时间求导才可以如此简略表示。
r cosine theta is a product of two quantities. By the way, we introduce a notation, r dot. Dot is equal to d dt. This condition is very common in physics. Everybody uses this dot notation as a derivative. So we are now calculating r dot.(r乘以cosθ是两个量的乘积。顺便说一下，我们引入一个符号，r点。点等于d/dt。这个条件在物理学中非常常见。大家都用这个点符号表示导数。所以我们现在计算r点。)
The first one is r cosine theta dot. Now, does anybody know if I have two things, if I have a product, alpha and beta, which is a time derivative. There is a rule for this. This rule is very simple. You must remember it. Namely, this is very simple.(第一个是r乘以cosθ的导数。现在，有人知道吗，如果我有两个量，比如α和β的乘积，再求时间导数。这里有一个规则。这个规则很简单。你们必须记住它。也就是说，这非常简单。)
乘法求导法则： ( α β ) ′ = α β ′ + α ′ β (\alpha \beta)'=\alpha \beta' + \alpha' \beta (αβ)′=αβ′+α′β
Alpha dot beta plus alpha beta dot. If you have not learned this, you will be learning this soon. It is very easy to prove. But anyway, remember this. So now we are going to calculate this. So r cosine theta, the time derivative would consist of two terms.(α'乘以β加上α乘以β'。如果你还没学过这个，你很快就会学到。这很容易证明。但无论如何，记住这一点。现在我们要来计算这个。所以r乘以cosθ，其时间导数将包含两项。)
First, I take r dot. Second, I take cosine theta dot. So the first term is r dot cosine theta. The second term is the derivative of cosine theta. Now I use what is called a chain rule. The chain rule is d cosine theta dt is equal to d cosine theta d theta times d theta dt.(首先，我取r点。其次，我取cosθ点。所以第一项是r点乘以cosθ。第二项是cosθ的导数。现在我使用所谓的链式法则。链式法则就是d(cosθ)/dt等于d(cosθ)/dθ乘以dθ/dt。)
v ⃗ = d r ⃗ d t = r ⃗ ˙ = ( r ˙ cos θ − r sin θ θ ˙ , r ˙ sin θ + r cos θ θ ˙ ) \vec{v}=\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}=\dot{\vec{r}}=(\dot{r}\cos \theta-r\sin \theta \dot{\theta}, \dot{r}\sin \theta+r\cos \theta\dot{\theta}) v =dtdr =r ˙=(r˙cosθ−rsinθθ˙,r˙sinθ+rcosθθ˙)
d theta dt is theta dot. This is called the chain rule. It is another fundamental formula. Okay. So the first term is r dot cosine theta. The second term is r times cosine theta with respect to theta, the derivative. And that is minus sine theta. So minus r sine theta times theta dot. This is just the x-coordinate. It is just this part. The y-coordinate, we do the same. It becomes r dot sine theta plus r cosine theta theta dot.(dθ/dt就是θ点。这被称为链式法则。它是另一个基本公式。好的。第一项是r点乘以cosθ。第二项是r乘以cosθ对θ的导数，也就是负sinθ。所以是负r乘以sinθ再乘以θ点。这只是x坐标。就是这部分。y坐标，我们做同样的处理。它变成了r点乘以sinθ加上r乘以cosθ乘以θ点。)
Because the derivative of sine theta is cosine theta. Okay. Finally, we need to take the derivative again. Because we need acceleration. So r double dot. This is acceleration. Now it is complicated. Let's look at the first term first. We have four terms here.(因为sinθ的导数是cosθ。好。最后，我们需要再次求导。因为我们需要加速度。所以是r的双点。这就是加速度。现在变得复杂了。我们先来看第一项。这里共有四项。)
r ⃗ ¨ = ( r ¨ cos θ − r ˙ sin θ θ ˙ − r ˙ sin θ θ ˙ − r cos θ θ ˙ 2 − r sin θ θ ¨ , r ¨ sin θ + r ˙ cos θ θ ˙ + r ˙ cos θ θ ˙ − r sin θ θ ˙ 2 + r cos θ θ ¨ ) = [ ( r ¨ − r θ ˙ 2 ) cos θ − ( 2 r ˙ θ ˙ + r θ ¨ ) sin θ , ( r ¨ − r θ ˙ 2 ) sin θ + ( 2 r ˙ θ ˙ + r θ ¨ ) cos θ ] \begin{align} \ddot{\vec{r}}&=(\ddot{r}\cos \theta-\dot{r}\sin \theta \dot{\theta}-\dot{r}\sin \theta \dot{\theta}-r\cos \theta \dot{\theta}^2-r\sin \theta \ddot{\theta}, \quad\ddot{r}\sin \theta+\dot{r}\cos \theta \dot{\theta}+\dot{r}\cos \theta\dot{\theta}-r\sin \theta \dot{\theta}^2+r\cos \theta \ddot{\theta}) \\[1.5ex] &=[(\ddot{r} - r\dot{\theta}^2)\cos\theta - (2\dot{r}\dot{\theta} + r\ddot{\theta})\sin\theta, \quad(\ddot{r} - r\dot{\theta}^2)\sin\theta + (2\dot{r}\dot{\theta} + r\ddot{\theta})\cos\theta] \end{align} r ¨=(r¨cosθ−r˙sinθθ˙−r˙sinθθ˙−rcosθθ˙2−rsinθθ¨,r¨sinθ+r˙cosθθ˙+r˙cosθθ˙−rsinθθ˙2+rcosθθ¨)=[(r¨−rθ˙2)cosθ−(2r˙θ˙+rθ¨)sinθ,(r¨−rθ˙2)sinθ+(2r˙θ˙+rθ¨)cosθ]
One term, two terms, three terms , four terms. Let's look at them one by one. The first term is r dot cosine theta. So according to the rule just now, the first term is r double dot cosine theta. The second term is r dot times cosine theta derivative. So it is minus r dot sine theta theta dot.
This is the first term. The second term is more complicated. The second term is r sine theta theta dot. So you have three terms here. The first term is r double dot sine theta theta dot. The second term is minus r cosine theta theta dot.
So it is theta dot square. The third term is minus r sine theta theta double dot. So there are five terms in the x coordinate acceleration.
Do you agree with these five terms? And then look at the y coordinate. Also, there are five terms. r double dot sine theta. Plus r dot cosine theta theta dot. Plus r dot sine theta theta dot. Plus r cosine theta theta dot squared.(以上内容，直接见公式，注：视频中截止当前进度计算结果有误，杨老后面纠正)
加上RcosθθRd, 一二三四五也是五项,所以x-coordinate有五项,y-coordinate有五项,这个看起来很复杂,是很复杂的。 不过呢,你仔细看看,你看这个x有一个Rd,乘上cosθ,y有一个Rd,sinθ,
The direction of cosθ and sinθ is the same as the direction of r So we can write these two terms together Objective dislocation is called r hat r hat is the unit vector in the direction of r So what is r hat?（cosθ和sinθ的方向与r的方向相同，因此我们可以将这两项合并写在一起。目标位错称为r帽，r帽是r方向上的单位向量。那么，什么是r帽？）
r ⃗ \vec{r} r 方向上的单位向量： r ^ = ( cos θ , sin θ ) = r ⃗ r \hat{r}=(\cos \theta, \sin \theta)=\frac{\vec{r}}{r} r^=(cosθ,sinθ)=rr
r hat is cosθ and sinθ which is equal to r divided by r Then we have sinθ sinθ is the point here which is in the direction of r So r hat is in this direction sinθ is sinθ which is increasing towards sinθ So it is not proportional to r This is sinθ sinθ is perpendicular to r So r hat is perpendicular（r帽等于cosθ和sinθ，也就是r除以r。然后我们有sinθ，sinθ在这里的点是沿着r方向的。所以r帽在这个方向上，sinθ是sinθ，它朝着sinθ增加。所以它与r不成正比。这是sinθ，sinθ垂直于r）
θ ^ ⊥ r ^ , θ ^ = ( − sin θ , cos θ ) \hat{\theta} \perp \hat{r}, \quad \hat{\theta}=(-\sin \theta, \cos \theta) θ^⊥r^,θ^=(−sinθ,cosθ)
So what do you think of theta hat? It is minus sinθ cosθ Think about it and you will know that this is right And this is right, no matter how much θ is In this case, θ is less than 90 degrees If θ is 120 degrees, this formula is still right This is the beauty of mathematics If you get the formula right, the general case is always right Alright, let's look at the second example This is r dot theta dot (那么你觉得θ帽怎么样？它是负sinθ cosθ。仔细想想你就会知道这是对的。而且无论θ有多大，这个公式都是正确的。在这个例子中，θ小于90度。如果θ是120度，这个公式仍然成立。这就是数学的美妙之处。如果你得到了正确的公式，那么普遍情况总是成立的。好了，我们来看第二个例子。这是r点θ点。)
r ⃗ ¨ = ( r ¨ cos θ − r ˙ sin θ θ ˙ − r ˙ sin θ θ ˙ − r cos θ θ ˙ 2 − r sin θ θ ¨ , r ¨ sin θ + r ˙ cos θ θ ˙ + r ˙ cos θ θ ˙ − r sin θ θ ˙ 2 + r cos θ θ ¨ ) = [ ( r ¨ − r θ ˙ 2 ) cos θ − ( 2 r ˙ θ ˙ + r θ ¨ ) sin θ , ( r ¨ − r θ ˙ 2 ) sin θ + ( 2 r ˙ θ ˙ + r θ ¨ ) cos θ ] = r ⃗ ¨ r ^ + 2 r ˙ θ ˙ θ ^ − r θ ˙ 2 r ^ + r θ ¨ θ ^ = ( r ¨ − r θ ˙ 2 ) r ^ + ( 2 r ˙ θ ˙ + r θ ¨ ) θ ^ \begin{align} \ddot{\vec{r}}&=(\ddot{r}\cos \theta-\dot{r}\sin \theta \dot{\theta}-\dot{r}\sin \theta \dot{\theta}-r\cos \theta \dot{\theta}^2-r\sin \theta \ddot{\theta}, \quad\ddot{r}\sin \theta+\dot{r}\cos \theta \dot{\theta}+\dot{r}\cos \theta\dot{\theta}-r\sin \theta \dot{\theta}^2+r\cos \theta \ddot{\theta}) \\[1.5ex] &=[(\ddot{r} - r\dot{\theta}^2)\cos\theta - (2\dot{r}\dot{\theta} + r\ddot{\theta})\sin\theta, \quad(\ddot{r} - r\dot{\theta}^2)\sin\theta + (2\dot{r}\dot{\theta} + r\ddot{\theta})\cos\theta] \\[1.5ex] &=\ddot{\vec{r}}\hat{r}+2\dot{r}\dot{\theta}\hat{\theta}-r\dot{\theta}^2\hat{r}+r \ddot{\theta}\hat{\theta} \\[1.5ex] &=(\ddot{r}-r \dot{\theta}^2)\hat{r} + (2\dot{r}\dot{\theta}+r \ddot{\theta})\hat{\theta} \end{align} r ¨=(r¨cosθ−r˙sinθθ˙−r˙sinθθ˙−rcosθθ˙2−rsinθθ¨,r¨sinθ+r˙cosθθ˙+r˙cosθθ˙−rsinθθ˙2+rcosθθ¨)=[(r¨−rθ˙2)cosθ−(2r˙θ˙+rθ¨)sinθ,(r¨−rθ˙2)sinθ+(2r˙θ˙+rθ¨)cosθ]=r ¨r^+2r˙θ˙θ^−rθ˙2r^+rθ¨θ^=(r¨−rθ˙2)r^+(2r˙θ˙+rθ¨)θ^
This is r dot theta dot This is cosθ which is equal to theta hat So it becomes r dot theta dot times theta hat The third one is minus sinθ This is r dot sinθ r dot This is r dot cosθ r sinθ This is r dot cosθ This is right This is r double dot r double dot This should be one dot r dot sinθ theta dot Yes, this is r dot cosθ So this is plus r dot theta dot theta hat You have to do this carefully And this is an exercise So if I make a mistake in this calculation You have to do it yourself The last one is this one This is r cosθ minus r This is right This one is This is sin And this is minus So it's minus r theta dot squared r hat The last one is plus r theta double dot theta hat Five items Let me see Is this the same as what I calculated last night? Yes You can take a look at this The last two items are the same So it becomes r double dot plus two times r dot theta dot theta hat Minus r theta dot squared r hat Plus r theta double dot theta hat This is r hat There are two items for r hat and two items for theta hat So take out r hat Take out r hat Multiply by r double dot r double dot minus r theta dot squared Plus theta hat Theta hat is two times r dot theta dot Plus r theta double dot You didn't know that, did you? This is the last item There are two items for r One is for r hat One is for theta hat There are two items for r r double dot minus r theta dot squared In the direction of theta It is two times r dot theta dot plus r theta double dot (不翻译了，直接看上面公式)
This formula is very important Why is it very important? Newton wrote the Principia Mathematica He used this formula As I told you before He invented differential geometry Differential calculus It is obvious that he had this formula Through differentiation Just like we did But in the book He did not present it in this fashion He presented it by a geometrical argument Which is much much more difficult (这个公式非常重要。为什么它如此重要？牛顿撰写了《自然哲学的数学原理》，他运用了这个公式。正如我之前告诉你的那样，他发明了微分几何和微分学。显然，他掌握了这个公式。通过微分运算，就像我们所做的那样。但在书中，他并没有以这种方式呈现，而是通过几何论证来展示，这要困难得多。)
Today we don't follow the geometrical argument There is no need to follow the geometrical argument We follow the argument that we just made And this formula From this formula He derived Kepler's three laws About the planetary motion And that was such a stunning achievement It gave birth to modern science（今天我们不再遵循几何论证 无需遵循几何论证 我们遵循刚刚提出的论证 从这个公式中 他推导出了开普勒三大定律 关于行星运动的规律 这一成就令人惊叹 它催生了现代科学）
So this is a very very important formula We are going to come back to it later Concentrated derivation In many senses it is very simple It is just to follow a few rules of differentiation But I advise each of you To go through the individual steps And understand the individual steps And anyway that is going to be one of the exercises （所以这是一个非常重要的公式，我们稍后会再回到这个公式。集中推导在很多方面都很简单，只需要遵循一些微分规则。但我建议你们每个人都亲自过一遍每一个步骤，并理解每一个步骤。无论如何，这将成为练习之一。）
特殊平面运动------匀速圆周运动
For next week Don't tell me that I am taking a special case The special case is uniform circular motion Uniform circular motion Uniform circular motion R is fixed So R dot is equal to zero C dot is omega It is equal to a constant It is a constant So C double dot is equal to zero Then what is R under this condition? （下周别跟我说我在讨论特殊情况 特殊情况就是匀速圆周运动 匀速圆周运动 匀速圆周运动 r 是固定的 所以 r ˙ \dot{r} r˙ 等于零 θ ˙ \dot{\theta} θ˙等于ω 它是一个常数 是一个常数 所以 θ ¨ \ddot{\theta} θ¨ 等于零 那么在这种情况下 R 是什么？）
在匀速圆周运动中， r r r 是固定的，且角速度恒定，所以有：
r ˙ = 0 θ ˙ = ω θ ¨ = 0 \dot{r}=0 \quad \dot{\theta}=\omega \quad \ddot{\theta}=0 r˙=0θ˙=ωθ¨=0
代入前面的公式中： r ⃗ ¨ = ( r ¨ − r θ ˙ 2 ) r ^ + ( 2 r ˙ θ ˙ + r θ ¨ ) θ ^ = − r ω 2 r ^ = − r ( v 2 r 2 ) r ^ = − v 2 r r ^ \begin{align} \ddot{\vec{r}}&=(\ddot{r}-r \dot{\theta}^2)\hat{r} + (2\dot{r}\dot{\theta}+r \ddot{\theta})\hat{\theta} \\[1.5ex] &=-r\omega^2\hat{r} = -r\left( \frac{v^2}{r^2} \right)\hat{r} \\[1.5ex] &=-\frac{v^2}{r}\hat{r} \end{align} r ¨=(r¨−rθ˙2)r^+(2r˙θ˙+rθ¨)θ^=−rω2r^=−r(r2v2)r^=−rv2r^
r ^ \hat{r} r^ 的方向是离心的，前面符号表明，匀速圆周运动的加速度方向是向心的，大小与我们之前推导的一致，且没有切向加速度分量。
R double dot is equal to R hat Multiply R dot is equal to zero So R double dot is equal to zero Minus R theta dot square So it is minus R omega square R dot is equal to zero Theta double dot is equal to zero So the last term is gone So what is this? This is equal to R This is minus R Omega square is V square over R square So it is equal to minus R hat Multiply by V square over R That is what we had last time （直接看上述公式，结论与之前推导的一致）
The magnitude is V square over R The direction is positive This calculation can also The calculation just now can also Make us think that our formula is correct Ok, any questions? For the two differentiation formulas I used just now One is the derivative of alpha beta It is equal to the sum of two terms（大小等于V平方除以R，方向为正。这个计算也可以...刚才的计算也让我们认为我们的公式是正确的。好的，有什么问题吗？关于我刚才使用的两个微分公式，一个是αβ的导数，它等于两项的和。）
How many people have not learned that? Anybody has learned that? I think If you have learned that, raise your hand How come? How many people don't know this formula? Then this is wrong Ok, I assume that you actually all have learned it Now how about the other formula? The derivative of f with respect to x Is equal to the derivative of f with respect to y Times dy dx How many people know this formula? （有多少人没学过这个？有人学过吗？我觉得如果你学过，请举手。怎么会这样？有多少人不知道这个公式？那就不对了。好吧，我假设你们其实都学过。那另一个公式呢？f对x的导数等于f对y的导数乘以dy/dx。有多少人知道这个公式？）
How many people don't know this formula? I think the interpretation is probably Some people are not quite sure Whether they know this formula Ok, if you are not sure A. You now know this formula You now have heard of this formula B. Try to review and learn this formula These are two fundamental rules You have to learn them If you do not know either of these two rules Intuitively You cannot become a physicist （有多少人不知道这个公式？我觉得解释可能是有些人不太确定他们是否知道这个公式。好吧，如果你不确定的话：A. 你现在知道了这个公式，你现在听说过这个公式了；B. 试着复习并学习这个公式。这是两条基本规则，你必须学会它们。如果你连这两条规则都不知道，直觉上，你不可能成为一名物理学家）
第五章：牛顿定律
牛一定律
Ok, next we are going to go to chapter 5 Chapter 5 is mainly about The elementary, the fundamental laws of Newton First law The first law is that If there is no force acting on it It will not move It will maintain its velocity There is no acceleration Why is this law not so obvious? Because there is friction in general（好的，接下来我们将进入第五章。第五章主要讲述牛顿的基本定律，即第一定律。第一定律指出：如果一个物体不受外力作用，它将保持静止或匀速直线运动状态，不会产生加速度。为什么这条定律看起来不那么显而易见呢？因为现实中普遍存在摩擦力。）
伽利略
So when something moves on a table Or if you shoot a projectile It will eventually stop Because of friction And in early years People did not understand, did not realize That there is air And air can cause resistance It was only later That it became understood by physicists Actually it was not Newton who first realized this It was Galileo Galileo was born in 1564 He died in 1642（因此，当某物在桌上移动或你发射一个抛射物时，它最终会停下来，这都是因为摩擦力的作用。而在早期，人们并不理解，也没有意识到空气的存在以及空气会产生阻力。直到后来，物理学家们才逐渐明白了这一点。实际上，最早意识到这一点的并非牛顿，而是伽利略。伽利略出生于1564年，逝世于1642年。）
A very interesting accident Is that the year that Galileo died Newton was born And Galileo already realized That without friction Without friction of forces A body would keep on moving And this today looks like a trivial conclusion But it was not trivial at all at that time But Galileo did not formulate it as a law Newton formulated it as a law （一个非常有趣的事实是，伽利略去世那年牛顿出生了。伽利略当时已经意识到，如果没有摩擦力，物体就会一直运动下去。这个结论在今天看来微不足道，但在当时绝非如此。不过伽利略没有将其表述为定律，是牛顿最终将其确立为定律的。）
So now we universally call it Newton's first law By the way if you are interested I recommend to you a book published in the last five years Called Galileo's Daughter I'm pretty sure the library has a copy of this And I'm pretty sure it has been translated into Chinese Galileo had two daughters He was very close to his first daughter（所以现在我们普遍称之为牛顿第一定律。顺便说一句，如果你感兴趣的话，我向你推荐一本近五年出版的书，名为《伽利略的女儿》。我很确定图书馆有这本书的副本，而且我相信它已经被翻译成中文了。伽利略有两个女儿，他与大女儿的关系非常亲密。）
Those daughters became nuns That was very common at that time And the correspondence between his first daughter and he Was preserved So a woman author in the United States Who was ethnically Italian Her parents were Italians Looked through these letters And wrote this book which became a very popular book It's very interesting to read it（那些女儿们成了修女，这在当时很常见。而他的大女儿与他之间的往来书信被保存了下来。于是，一位祖籍意大利的美国女作家------她的父母都是意大利人------翻阅了这些信件，并写出了这本后来非常受欢迎的书。读起来非常有趣。）
Because you then understand more About the style of life in those days You also understand more Galileo's struggle And you also understand more The relationship between father and daughter At the end of the book there was a great surprise I will not tell you what the surprise is Before that was revealed in the book。Physicists didn't know About something relating to the end of Galileo's life (因为这样你就能更了解那个时代的生活方式，也能更理解伽利略的斗争，以及父女之间的关系。书的最后有一个巨大的惊喜，我不会告诉你惊喜是什么，在书中揭示之前。物理学家们并不了解与伽利略生命终结相关的某些事情。)
!info
甚至在今天，在圣克罗切大教堂受万人瞻仰的伽利略墓上，没有任何碑文说明那里是修女玛丽亚·切莱斯特的埋骨之所。
但她的的确确就在那里。
可能很多人并不一定有时间认真阅读这本书，我来剧透一下，希望不要介意，这两段话是全书的最后两句话，指的是伽利略与女儿玛丽亚·切莱斯特修女合葬的感人细节。书中提到 18 世纪迁葬伽利略时，人们打开旧墓发现除了伽利略的棺木外，还有一具年轻女性的棺木，即伽利略最疼爱的长女玛丽亚·切莱斯特。这一发现揭示了父女之间超越生死的情感纽带，成为全书最动人的段落之一。
力与净力
According to the book, there is a discussion about force and net force. If there is a force on a body, then there is acceleration. That will come when we come to the second law.（根据书中的内容，有一段关于力和净力的讨论。如果一个物体受到力的作用，那么它就会产生加速度。这一点将在我们讨论第二定律时详细阐述。）
But if a body suffers several forces, then you first add those forces, and that's called the net force. So if a body suffers two forces, you take the sum, the vector sum of the forces, and that is the net force on that body. When we talk about the force law, I said that when a body does not receive force, it is a uniform motion, its speed does not change.（但如果一个物体受到多个力的作用，那么你首先要将这些力相加，这被称为合力。因此，如果一个物体受到两个力的作用，你需要取这两个力的矢量和，这就是作用在该物体上的合力。当我们讨论力的定律时，我说过，当一个物体不受力时，它会做匀速运动，其速度不会改变。）
This sentence has a premise. The premise is that the observer is in an inertial reference system. If a person is sitting in the elevator, then when he does measurements in the elevator, he does not meet this law.（这句话有一个前提。前提是观察者处于惯性参考系中。如果一个人坐在电梯里，那么当他在电梯内进行测量时，他就不符合这个定律。）
不要"钻牛角尖"
So there is a problem. How do you know that the experiment you are doing is in an inertial reference system? This problem is half philosophical and half physical. I think this is not something that should be discussed in early physics.（所以这里有一个问题。你怎么知道你正在做的实验是在一个惯性参考系中进行的？这个问题一半是哲学性的，一半是物理性的。我认为这不应该在早期物理学中讨论。）
So I think you simply think that the attitude in this book is that the definition of an inertial reference system is that Newton's first law is correct. This seems to be an argument that goes round and round. It is an argument that goes round and round. I have a very complicated question here. Don't ask too much. The day before yesterday, a classmate came to talk to me.（所以我认为你只是认为这本书中的观点是，惯性参考系的定义就是牛顿第一定律成立。这似乎是一个循环往复的争论。这是一个循环往复的争论。我这里有一个非常复杂的问题。不要问太多。前天，有个同学来找我谈话。）
I told him, I said, in the process of learning, don't be stubborn. This doesn't mean that you don't understand some things. You just think. You shouldn't think. That's not what I mean. If you don't understand something, you have to think about it.（我对他说，在学习的过程中，不要钻牛角尖。这并不意味着你对某些事情不理解。你就只是想，你不该去思考。这不是我的意思。如果你不明白某件事，你必须去思考它。）
Just like what I said last time, if you feel that something is different from your intuition, this is the best chance to learn. But if you have been studying for three days and you haven't figured it out, then my advice is that there must be something very troublesome here. You'd better not be stubborn in this regard.（正如我上次所说，如果你觉得某些事情与你的直觉不符，这正是学习的最佳时机。但如果你已经研究了三天还没搞明白，那么我的建议是这里肯定有什么非常棘手的问题。在这方面你最好不要钻牛角尖。）
When you have more knowledge, sometimes you feel that your original view is not necessary. In other words, it is good to be a little stubborn. It is not good to be stubborn all the time. （当你拥有更多知识时，有时会觉得原来的观点并非必要。换句话说，有点固执是好的，但一直固执就不好了。）
牛二定律
F ⃗ net = m a ⃗ \vec{F}_{\text{net}}=m \vec{a} F net=ma
Next, we are going to talk about the second law. The simplest second law is F net is equal to m a. We have already discussed m. We also know the unit. We have also discussed a. We also know the unit.（接下来，我们将讨论第二定律。最简单的第二定律表达式是F net等于m a。我们已经讨论过m的含义，也知道其单位。同样，我们也探讨过a的含义，并了解其单位。）
So we have a unit of force called Newton. Newton's force is equal to one kilogram multiplied by one meter per second squared acceleration. If there is a kilogram of mass, you give it an acceleration.（所以我们有一个力的单位叫做牛顿。牛顿的力等于一千克乘以每秒平方的加速度。如果有一千克的质量，给它一个加速度。）
It is one meter per second squared. The force you need is one Newton. When I was a student, we did not use this. We used the CGS system. It is called a dyne. So what is the relationship between a Newton and a dyne? It is like this.（加速度为每秒一米平方。所需的力为一牛顿。我读书时并不用这个单位，而是用厘米-克-秒制（CGS单位制），力的单位叫达因。那么牛顿和达因之间是什么关系呢？是这样的。）
You write one kilogram divided by a gram. Because the CGS system uses grams. Multiply by grams. This is the same as this. Then you write it as one meter divided by centimeters. Multiply by centimeters. This is also the same as meters. It is divided by second squared. Kilogram divided by grams is ten to the third power.(你写的是1千克除以1克。因为CGS制用的是克。乘以克。这和这个是一样的。然后你写成1米除以厘米。乘以厘米。这也和米一样。除以秒的平方。千克除以克等于10的3次方。)
Meter divided by centimeters is ten to the second power. So it becomes gram centimeters divided by second squared. This is a dyne. So it is ten to the fifth power. This means that a dyne is a very small force. This makes sense.(米除以厘米是十的二次方。所以变成了克厘米除以秒的平方。这就是达因。所以是十的五次方。这意味着达因是一个非常小的力。这是有道理的。)
1 N = ( 1 kg ) ( 1 m/s 2 ) = ( 1 kg g ) g ( 1 m cm ) cm / s 2 = 1 0 5 dyne 1\, \text{N}=(1\,\text{kg})(1\,\text{m/s}^2)=\left( \frac{1\,\text{kg}}{\text{g}} \right)\text{g}\left( \frac{1\,\text{m}}{\text{cm}} \right)\text{cm} / \text{s}^2=10^5\,\text{dyne} 1N=(1kg)(1m/s2)=(g1kg)g(cm1m)cm/s2=105dyne
Because the force that Newton usually uses is more convenient to do with Newton. If you use a dyne, it becomes a big number. Of course it is more inconvenient. It is not convenient in daily life. By the way, when you do exercises, you can use Chinese or English. I am going to write research papers also in English.(因为牛顿通常使用的力用牛顿来计算更方便。如果用达因，数字会变得很大，当然就更不方便了。在日常生活中也不实用。顺便说一下，做练习时可以用中文或英文。我写研究论文也会用英文。)
But I will accept handwritings which are in Chinese. We know that a barbell is attracted by the gravity of the earth. So it has to fall. Then it has an acceleration. This acceleration is downward. But the magnitude is g. So the gravity of the earth is g. So the force of gravity is equal to the mass times g. So any body which is attracted by the gravity of the earth is mg.(但我可以接受手写的中文。我们知道杠铃受到地球引力的吸引。所以它必须下落。于是它就有了一个加速度。这个加速度是向下的。但大小是 g。所以地球的引力就是 g。所以重力等于质量乘以g。因此任何受地球引力作用的物体其重力都是 mg。)
g is 9.78 meters per second squared. There is a confusion about this. It is very common. It is called the concept of weight. The definition of weight is a force. It is not a mass. Technically, weight is a force. So what is the weight of a kilogram? It is a kilogram times g, which is 9.78. The final result is Newton. So the mass of a kilogram is 9.78 Newton.（重力加速度g为9.78米每二次方秒。对此存在一个常见的混淆概念------重量。严格来说，重量是一种力而非质量。那么一千克的重量是多少呢？它等于一千克乘以g（即9.78），最终结果以牛顿为单位。因此，一千克的质量对应9.78牛顿的力。）
But sometimes the weight is confused with the mass. For example, you ask a person how heavy he is. You ask a question. Of course, his answer should be force. But his answer is 50 kilograms. Yes, 50 kilograms is a mass. So you have to pay attention to this. Sometimes force is a mass times weight unit. And weight times mass unit.（但有时重量会与质量混淆。例如，你问一个人他有多重。你问的是一个力的问题。当然，他的回答应该是力。但他的回答是50公斤。是的，50公斤是质量。所以你必须注意这一点。有时力是质量乘以重量单位。而重量乘以质量单位。）
These two are often confused. So you have to figure out the difference. We say that anything is attracted by the gravity of the earth at any time. The force is mg. M is its mass. If something is not moving on the table, it has no acceleration. Without acceleration, it has no force. But I didn't say that the earth has a force. Why doesn't it move? It must be because the table gives it a force.（这两个概念经常被混淆。所以你必须弄清楚它们的区别。我们说任何物体在任何时候都会受到地球引力的吸引。这个力是 mg，其中 m 是物体的质量。如果某物在桌子上没有移动，它就没有加速度。没有加速度，就没有力。但我没说地球对它施加了力。那它为什么不动呢？肯定是因为桌子给了它一个力。）
So its net force is zero. But the net force is a combination of two things. One is mg. The image is the force of the earth. The other is the force of the table. The force of the table is called normal force. It is also mg. So these two are balanced. The net force is zero. So it doesn't move. Another point is tension. If this wood is here, this board is here.（所以它的净力为零。但净力是由两个因素组合而成的。一个是重力mg，即地球对物体的引力。另一个是桌面对物体的支撑力，这种支撑力被称为法向力，大小也是mg。因此这两个力相互平衡，净力为零，所以物体不会移动。另一个概念是张力。假设这块木板在这里，这块木板在这里。）
牛三定律
I pull it. When I pull it, I use a force. This force is called T. This is called tension. There is a tension on a rope. I think everyone knows this concept. Then there is a third law. The third law is that when two things interact, the force of A is equal to the force of B. The force of B is equal to the force of A. I think you all have learned this. And you will understand it in your daily life. It is also intuition.（我拉它。当我拉它时，我施加了一个力。这个力叫做 T。这被称为张力。绳子上存在张力。我想大家都知道这个概念。然后还有第三定律。第三定律是说，当两个物体相互作用时，A对B的力等于B对A的力。B对A的力等于A对B的力。我想你们都学过这个。在日常生活中你们也会理解它。这也是直觉。）
So if I look at this now. What do I use? I use a person. A person is pulling this. A person has a force like this for this block. So the force of the block is like this. But at the same time, the block pulls the person over there. So the two are opposite to each other. This is the third law. So the force on each rope is always two forces.（那么现在我来看这个。我该用什么？我用一个人。一个人正在拉这个。这个人对这个方块施加了一个这样的力。所以方块受到的力是这样的。但同时，方块也在那边拉这个人。所以两者是相互对立的。这就是第三定律。因此，每根绳子上的力总是两个力。）
F ⃗ A B = − F ⃗ B A \vec{F}{AB}=-\vec{F}{BA} F AB=−F BA
One is to pull the person on the left to the right. Pull the person on the right to the left. The opposite side is the opposite force. We can write this as FAB. The definition of FAB is On A by B. This is equal to negative FBA. On B by A. I think you all have learned these simple items in high school. Now I want you to study. You have just talked about these things. You are waiting now. Study Chapter 5. Review and Summary. You have all learned these simple things just now. Is there any problem? No. If there is no problem, let's do two exercises. （一种方法是将左边的人向右拉，将右边的人向左拉。相反的一侧则是相反的作用力。我们可以将其表示为FAB。FAB的定义是"由B作用于A的力"，这等于负的FBA，即"由A作用于B的力"。我想你们在高中时都学过这些简单的内容。现在我希望你们学习一下。你们刚刚讨论过这些东西。现在你们正在等待。学习第五章。复习与总结。你们刚才都学了这些简单的内容。有什么问题吗？没有。)
练习题
Let's say someone is standing in an elevator. The elevator is going up. There is an acceleration A. I want to know how much force the floor of the elevator gives to that person. How much force does the floor of the elevator give to that person? This is very simple. Because this person is here. He receives two forces. One force is MG. The other force is the force given by the floor.(如果没有问题，我们来做两个练习。假设有个人站在电梯里。电梯正在上升。有一个加速度 a。我想知道电梯地板给那个人施加了多少力。电梯地板给那个人施加了多少力？这很简单。因为这个人在这里。他受到两个力。一个力是 mg。另一个力是地板施加的力。)
We call it F. So what is the net force? It is equal to F minus MG. The F must be bigger than MG. If F is smaller than MG, the elevator will go down. Now we assume that the elevator is going up. This is equal to MA. So we know that F is equal to MG plus MA.(我们称之为F。那么净力是多少呢？它等于F减去MG。F必须大于MG。如果F小于MG，电梯就会下降。现在我们假设电梯正在上升。这等于MA。所以我们知道F等于MG加上MA。)
I will tell you a more complicated one. Let's say there is a train. There are many boxes. There are four boxes. They are placed on a smooth surface. Then I pull the box. I have a force to pull. If the force is F, and the mass of each box is M, then the net force is the internal force. Don't worry about the internal force.(我会告诉你一个更复杂的情况。假设有一列火车。里面有很多箱子。 这里有四个盒子。它们被放置在一个光滑的表面上。然后我拉动盒子。我有一个拉力。如果拉力是F，每个盒子的质量是m，那么净力就是内力。不用担心内力。)
For example, a block of wood. When I pull it, there is a force between the front and the back of the wood. If you cut the wood in half, there is a force between the front and the back of the wood. The force is because of the relationship between the front and the back of the wood. When the force is small, don't worry about it. It is the same as this.(例如一块木头。当我拉动它时，木头的前后之间会产生一个力。如果你把木头切成两半，木头的前后之间依然存在这个力。这个力源于木头前后部分之间的关联。当这个力很小时，不必担心。道理与此相同。)
The first block is pulled by the second block. The second block is pulled by the first block. These are small. These are internal forces. Don't worry about it. So what is the final result? Let's say there is an acceleration A. This A is the mass. Its mass is 4M. So the force is 4MA. Is there any problem with this equation? No problem.(第一个块被第二个块拉动。第二个块被第一个块拉动。这些都是小的。这些都是内力。不用担心。那么最终结果是什么？假设有一个加速度A。这个A是质量。它的质量是4M。所以力是4MA。这个方程有什么问题吗？没有问题。)
Then I ask about the first mass. The first one. This is the total. Let's look at the first equation. The first block. The first block has two forces. One is F. One is the tension. The first tension. I call it T1. T1 is the tension of the second block. So F-T1 is the net force. The net force is MA. So what is T1? T1-F-MA is 3MA. Similarly, this is 2MA. This is 1MA.(然后我问第一个质量。第一个。这是总数。我们来看第一个方程。第一个块。第一个块有两个力。一个是F。一个是张力。第一个张力。我叫它T1。T1是第二个块的张力。所以F-T1是净力。净力是MA。那么T1是什么？T1-F-MA是3MA。同样，这是2MA。这是1MA。)
共 4 节车厢，总质量为 4m，加速度为 a，则第一节车厢与第二节车厢之间的张力 T 1 = F − m a = 3 m a T_{1}=F-ma=3ma T1=F−ma=3ma，同理第二节和第三节车厢之间张力 T 2 = 2 m a T_{2}=2ma T2=2ma，第三节和第四节车厢之间张力 T 3 = m a T_{3}=ma T3=ma，所以如果你拖拽，这几个车厢之间最可能断开的就是在头部位置，因为头部车厢之间受到的拉力最大。
In fact, let's say it is a very big block. If in that block, the block is not divided like this. In fact, when you pull it, one-fourth of the block is pulled by the three behind. Then there is a tension. That tension is three-fourths of the original tension. Half of it is the half of that tension.(事实上，我们可以说这是一个非常大的块体。如果在这个块体中，块体没有被这样分割。实际上，当你拉动它时，块体的四分之一被后面的三个部分拉动。这样就产生了张力。这个张力是原始张力的四分之三。其中一半就是那个张力的一半。)
When you pull it to the end, the tension becomes smaller. So there is tension in a block. In other words, if it is not a block, but something that is easy to break, then when you pull it, the broken part will be cut off. It must be the part above the head. Why? Because you have to pull a lot of things above the head. So it is heavier. It is heavier. It is powerful. If you think about it carefully, it is a very natural thing.(当你拉到尽头时，张力会变小。因此，方块内部存在张力。换句话说，如果不是方块，而是容易断裂的东西，那么当你拉动时，断裂的部分会被切断。断裂的一定是在拖拽的头部。为什么呢？因为你必须拉动头部的许多东西。所以它更重。它更重了。它更有力量。仔细想想，这是很自然的事情。)
五次讲座课程小结
We have been teaching for two weeks. First, everyone, as for the situation of the mixture of English and Chinese that I am talking about now, is it 50% or 80% or 95%? Raise your hand if you think it is 50%. Don't be shy. Raise your hand if you think it is 75%. Raise your hand if you think it is 95%. Raise your hand if you think it is 25%.(我们已经教了两周了。首先，各位，关于我现在说的中英文混合的情况，是50%还是80%还是95%？认为50%的请举手。别害羞。认为75%的请举手。认为95%的请举手。认为25%的请举手。)
I think a lot of people don't want to raise their hands. But my conclusion is that I think everyone can do it. Basically, basically, everyone can accept I hope in future weeks, we can even increase the percentage of English in my language. The second problem is the speed of our progress now. After the first lecture, some people... Because I usually don't look at the Internet. I think everything on the Internet is just noise.(我认为很多人不愿意举手发言。但我的结论是，我觉得每个人都能做到。基本上，基本上每个人都能接受。我希望在接下来的几周里，我们甚至能提高我语言中英语的比例。第二个问题是我们目前的进展速度。在第一节课后，有些人......因为我通常不看网络。我觉得网上的东西都是噪音。)
But a professor, a physics professor, forwarded some comments written on the Internet to me. I read it. I got the impression that most people think it is too shallow. Do you still think it is too shallow now? If anyone has a strong opinion on this, you can talk to me from 4 to 5 p.m. next Monday. If you don't have a strong opinion, this is what I will do in the future. This is what I told you in advance.(但有一位教授，一位物理学教授，把网上的一些评论转发给了我。我看了。我的印象是，大多数人认为这太浅显了。你现在还觉得太浅显吗？如果有人对此有强烈的看法，下周一下午4点到5点可以和我谈谈。如果你没有强烈的意见，这就是我以后要做的事情。这是我提前告诉你的。)
网上的评论认为杨老这五次的内容比较简单，我猜可能有这么几个方面，第一是前 5 章内容本身简单，大部分都是中学已经学过的，只有少量的最基础的一些微分或积分操作；其次能参与学习的网友本身具备一定高度的物理知识储量，而这本身是面向大一的普通物理，因此感觉过于基础；还有可能是认为杨老作为一代物理宗师，可能会从理论物理高度讲起，谈一些较为高深的知识体系。
I will essentially go through the book, but add some material which is not in the book. However, some of the exercises will include these additional topics. But for the two examinations, I will only use material in the book.(我基本上会按照书上的内容来讲，但会补充一些书中没有的材料。不过，部分练习会涉及这些额外的主题。但两次考试的内容将仅限于书中的材料。)
Let me remind everyone what I said the first time. I advise you in order to improve your English, not only your pronunciation, every aspect of your English, to read aloud every day for maybe 3 or 5 minutes any paragraph of English. Open your mouth wide, and I guarantee if you do that for a month, you would observe yourself that you have greatly improved.(让我提醒大家我最初说过的话。我建议你们为了提高英语水平，不仅仅是发音，而是英语的各个方面，每天大声朗读3到5分钟的任何一段英文。张大嘴巴，我保证如果你坚持一个月，你会发现自己有了很大的进步。)
Okay. I think we are going faster than what I was originally told to. Originally, I was told that we should finish one and a half chapters per week. Now in two weeks, we finish two and a half chapters. I will observe how you are doing from your exercises, and adjust the speed. It turns out that many of you have great problems.(好的。我觉得我们的进度比我最初被告知的要快。最初，我被告知每周应该完成一章半的内容。现在两周内，我们完成了两章半。我会通过你们的练习来观察你们的表现，并调整进度。结果发现你们很多人存在很大的问题。)
But besides English, at the beginning, I think physics, according to the book, is quite easy. So you should spend more time improving your English and improving your calculus. Okay. We have 25 more minutes. Anybody can ask me questions? The formal class is over. Okay, you can ask this way.(但除了英语，一开始我觉得物理，根据书上的内容，是相当简单的。所以你应该花更多时间提高你的英语和微积分。好的，我们还有25分钟。有人可以问我问题吗？正式课程结束了。好的，你可以这样问。)
力的定义
学生问：牛二定律中的公式 F = m a F=ma F=ma 在有些书中是说，这是力的定义，不是定律的定义，说是在这个定律之前，还没有力的定义，所以，到底力的定义是什么？
The basic concepts like this become more and more complicated as time goes by and his understanding becomes deeper and more complicated. There is a definition of force So when it comes to physics, when it comes to the end of the 20th century, we know a lot about force. So when it comes to the definition of force in the textbook, I don't think it is very interesting.(随着时间的推移，人们对这些基本概念的理解越来越深入、越来越复杂，这些概念本身也变得越来越复杂。日常生活中有一个关于力的定义。而在物理学中，特别是 20 世纪末的物理学时，我们对力已经有了很多了解。因此，当教科书里提到力的定义时，我觉得它并不是很有趣。)
Because we know much more than this. If I punch you, of course I have a force. But where does this force come from? We know that the reason is that the atoms in my fist interact with the atoms in your fist.(因为我们知道的远不止这些。如果我打你一拳，我当然施加了一个力。但这个力从何而来？我们知道，原因在于我拳头中的原子与你拳头中的原子发生了相互作用。)
How does this interact work? It is because there are electrons and nuclei in each atom. There is a force between the electrons and the nuclei. So when this force is combined with the force in your atom, there is a complication.(这种相互作用是如何运作的？这是因为每个原子中都有电子和原子核。电子和原子核之间存在一种力。所以当这种力与你原子中的力结合在一起时，就会出现一种复杂的现象。)
Do we understand this complication? We understand it very well. What does it mean to understand it very well? It means that we understand its most basic rules. This is called electromagnetism. This is the great contribution of Maxwell's equation. So how much do we know about this force? How much do we know about it? We can even understand the details of this force. And we can make very accurate calculations.(我们理解这种复杂性吗？我们非常理解。非常理解意味着什么？意味着我们理解其最基本的规则。这就是所谓的 电磁学。这是麦克斯韦方程的伟大贡献。那么我们对这种力了解多少呢？我们对它了解多少？我们甚至可以理解这种力的细节。而且我们可以进行非常精确的计算。)
But if we want to know how much force the interaction between the atoms in my fist and the atoms in your fist will produce, this is a... No one can calculate this. Can you imagine this? So, if you want to talk about our understanding of force, you know, now you have seen a lot of philosophical discussions. There is a discussion, I don't know what it's called in Chinese, it's called reductionism. What is reductionism in Chinese? The earliest meaning of reductionism is like this.（但如果我们想知道我拳头里的原子和你拳头里的原子相互作用会产生多大的力，这是...没人能计算出来。你能想象吗？所以，如果你想讨论我们对力的理解，你知道，现在你已经看到了很多哲学讨论。有一个讨论，我不知道它在中文里叫什么，它叫还原论。还原论在中文里是什么意思？还原论最早的含义是这样的。）
Some people say, for example, Steven Weinberg, a very famous physicist, he said that we should reduce our studies. What is reductionism? It means to ask the most basic questions. He said that now we can... He didn't say that, but he meant that we don't need to study chemistry anymore.（例如，著名物理学家史蒂文·温伯格等人认为，我们应该简化研究方向。什么是还原论？就是探究最根本的问题。他的言下之意是------虽然他没有明说------我们已无需再研究化学了。）
Because the whole chemistry can be reduced into Schrodinger equation. Does this make sense or not? It makes sense, but it doesn't make sense. What makes sense is that I think no one denies that the final method of chemistry is Schrodinger equation.（因为整个化学都可以归结为薛定谔方程。这说得通还是说不通？既说得通，也说不通。说得通的是，我认为没有人会否认化学的终极方法就是薛定谔方程。）
There are all kinds of reasons to believe this. So now all chemists believe this. But where does this make no sense? Does this mean that there is no chemistry anymore? This is completely absurd.（有各种理由相信这一点。所以现在所有的化学家都相信这一点。但这在哪里说不通呢？这是否意味着化学已经不存在了？这完全是荒谬的。）
Of course there is chemistry. If you write down this equation, it doesn't mean that you know the interaction between two atoms, especially two complex atoms, or two molecules. So if you ask me, I think this idea of reductionism is one kind of idea. But the future of knowledge in the world cannot be limited to this reductionism. So our understanding of power has become very deep. And we know the basic principle of the most complex force.（当然存在化学。如果你写下这个方程式，并不意味着你了解两个原子之间的相互作用，尤其是两个复杂的原子或两个分子之间的作用。所以如果你问我，我认为这种还原论的想法只是一种观点。但世界知识的未来不能局限于这种还原论。因此，我们对力的理解已经变得非常深刻。而且我们知道最复杂力的基本原理。）
I believe we all understand it now. As for the problems you encounter on a daily basis, if we can calculate it, we should be able to calculate it. But we know that this cannot be calculated. So there are still some, for example, you need fluid mechanics, you need elastic mechanics, there are a series of disciplines. So I think this is the most correct long-distance view. After this view, if you don't understand the whole equation, you have to use the dictionary to discuss the definition of history.（我想我们现在都明白了。至于你们日常遇到的问题，如果能计算出来，我们应该就能算出来。但我们知道这是无法计算的。所以还有一些，比如你需要流体力学，你需要弹性力学，还有一系列学科。因此，我认为这是最正确的长远观点。在这个观点之后，如果你不理解整个方程，就必须用字典来讨论历史的定义了。）
I don't think there is much content in this. So this kind of almost philosophical discussion in a very shallow discussion environment I don't think it's a useful discussion. Do you have any other questions? I can't hear you well. Please come closer. If you are far away, I can't hear you. （我不认为这里面有很多内容。所以在这种非常肤浅的讨论环境中进行这种近乎哲学的讨论，我不认为这是一个有用的讨论。 你还有其他问题吗？我听不太清楚你说的话。请靠近一点。如果你离得太远，我听不见你说话。。）
这种对哲学讨论的态度在费曼的讲座中也多次出现。
四种基本作用力
有同学问，四种基本作用力想大一统是怎么回事？
There is a concept of unifying all forces into a single force.There is a concept of unifying all forces into a single force. And there is a concept of unifying all forces into a single force. Now there are four forces in the air. Now there is a theory that four forces are unified into a single force. What is going on? Yes. This student asked, some people say there are four forces.（有一个将所有力统一为单一力的概念。有一种观点认为可以将所有力统一为单一的力。还有一种观点认为可以将所有力统一为单一的力。现在存在四种力。现在有一种理论认为四种力可以统一为单一的力。这是怎么回事？是的。这位学生问道，有人说存在四种力。）
Some people say that these four forces should be converted into a single force. What is going on? This is indeed a great achievement in physics in the 20th century On the one hand, it is a place that has not been done yet In the early years of human physics research There are all kinds of forces So I think in China, in the West It is very early to know the power of electricity （有人说这四种力应该被统一为一种力。这是怎么回事呢？这确实是 20 世纪物理学的一项伟大成就。一方面，这是一个尚未完成的领域。在人类物理学研究的早期，存在着各种各样的力。所以我认为，无论是在中国还是西方，人们很早就认识到了电的力量。）
There is magnetic force The earliest electricity, everyone knows You put a stick You rub it It can suck up small pieces of paper So this is static electricity China knew this very early China also knew the magnetic force very early The West also knew very early But it was in 1820 that it was found It turns out that electricity and magnetic force are not one thing It is a very closely related thing This is a big thing in history （有磁力 最早的电，大家都知道 你拿一根棍子 你摩擦它 它能吸起小纸片 所以这就是静电 中国很早就知道了 中国也很早就知道磁力 西方也很早就知道 但直到 1820 年才发现 原来电和磁力不是一回事 而是非常密切相关的东西 这是历史上的一件大事）
The person in this regard is called Ehrsted So after the electricity came out Many physicists went to study Study after study Maxwell's equation came out Maxwell's equation was finally written in 1865 This equation says that electricity and magnetic force are joined together It can't be separated You have to discuss the two together In order to know its final structure So this is called electromagnetic mechanics（在这方面做出贡献的人叫埃斯特德。所以电被发现后，许多物理学家都去研究。经过不断的研究，麦克斯韦方程诞生了。麦克斯韦方程最终在 1865 年问世。这个方程表明电力和磁力是结合在一起的，无法分开 。你必须将两者放在一起讨论，才能了解其最终结构。因此这被称为电磁力学。）
Electromagnetic mechanics is of course A very important part of physics in the future I think maybe Speaking of its importance Electromagnetic mechanics is the most important part of physics Because it affects all the technologies we see now Then came the end of the 19th century Until that time Until the end of the 19th century Only two kinds of force One is electromagnetic force One is external gravitational force（电磁力学当然是物理学中非常重要的一部分。我认为在未来，说到它的重要性，电磁力学可能是物理学中最重要的部分，因为它影响着我们现在看到的所有技术。然后到了 19 世纪末，直到那时，直到 19 世纪末，人们只知道两种力：一种是电磁力，一种是万有引力。）
As for the power of chemistry At the end of the 19th century Although it is not entirely clear But most people know Because the power of chemistry is actually the power of electromagnetic force There was no theory at that time But the direction is already clear But by the end of the 19th century There was a phenomenon of radiation This phenomenon of radiation It was soon understood It is not electromagnetic Nor is it external gravitational force It is another kind For example, at that time There was a very important thing （至于化学的力量 在 19 世纪末 虽然还不完全清楚 但大多数人都知道 因为化学的力量实际上是电磁力的力量 当时还没有理论 但方向已经很明确了 但到了 19 世纪末 出现了一种辐射现象 这种辐射现象 很快就被理解 它不是电磁的 也不是万有引力 它是另一种 例如，在当时 有一个非常重要的事情）
That is to say The earth The earth is constantly in heat Absorbed heat from the sun So if you take the balance of these two To calculate There was a phenomenon That is to say, there must be another kind of energy in the earth So this energy was not known at that time It was later discovered to be radioactive So at the end of the 19th century At the beginning of the 20th century After a series of such discussions（也就是说，地球一直在散热、并吸收来自太阳的热量。因此，如果从这两者的平衡来计算，就会发现有这样一种现象：即地球内部必然还存在另一种能源。这种能源在当时并不为人所知，后来才被发现是放射性。所以，在 19 世纪末到 20 世纪初，经过一系列这样的讨论后）
In addition to the discovery of Becquerel and Gulliver Until there is actually another kind of force This force is now called the interaction force So we know that there is electromagnetic force There is gravitational force We also know that there are some interaction forces This interaction force at the beginning I think it has a variety of interaction forces I don't know what it is It was not until the late 1940s that I learned Many of these interaction forces are almost the same （除了贝克勒尔和居里夫妇的发现之外，实际上还存在另一种力。这种力现在被称为弱相互作用力。因此我们知道有电磁力，有万有引力，我们还知道有一些弱相互作用力。起初认为这种弱相互作用力有多种形式，但不知道具体是什么。直到 20 世纪 40 年代末，才了解到这些弱相互作用力中有许多几乎是相同的。）
So the name of the interaction force It was only in the 1940s Just started Put all the weak forces together Called weak interaction force So between 1911 and 1913 It was found that there was a atomic nucleus in an atom This atomic nucleus is very small And the power in the atomic nucleus Very, very big Much bigger than electromagnetic force and interaction force So this one After many years of research （因此，弱相互作用力的名称直到 20 世纪 40 年代才开始被使用，将所有弱力统一称为弱相互作用力。而在 1911 年至 1913 年间，科学家们发现原子内部存在一个原子核。这个原子核非常小，但其中蕴含的力量却极其巨大，远超过电磁力和万有引力。经过多年的研究，这一发现逐渐被证实。）
So in the early 1950s This is when I was a student It is very clear There are basically four kinds of forces in the world Electromagnetic force, interaction force, weak force and strong force The strong force is the force in the atomic nucleus At that time, of course, we knew that there was an atomic bomb The atomic bomb is so powerful Because it uses strong force This is much greater than the remaining three forces （因此，在 20 世纪 50 年代初，那时我还是个学生。当时非常明确的是，世界上基本上存在四种力：电磁力、万有引力、弱力和强力。强力是原子核内的力。当然，那时我们已经知道原子弹的存在。原子弹之所以如此强大，正是因为它利用了强力。这种力远远超过其他三种力。）
So in the past 50 years We all know there are four kinds of forces So before this In 1915 and 1916 Einstein Einstein is definitely a genius He came up with a problem that no one else wanted to think of Because he said that the interaction force We all admit that Newton's interaction force But this is not good He integrated the interaction force He said that in fact, the so-called interaction force In fact, because space and time are not Euclidean It's not the integration of Euclidean It's an integration of the atomic nucleus （所以在过去的 50 年里，我们都知道有四种力。在此之前，在 1915 年和 1916 年，爱因斯坦------他绝对是个天才------提出了一个没人愿意思考的问题。因为他指出，万有引力------我们都承认牛顿的万有引力------但这并不理想。他将万有引力几何化，他说实际上所谓的万有引力，其实是因为时空并非欧几里得式的。它不是欧几里得几何的集合，而是弯曲的集合。）
This integration of the atomic nucleus, because no one understood So he thought he had an interaction force So it's equal to his general relativity What Einstein did in 1915 and 1916 Is to integrate the interaction force I think this is a great invention in human history So after that Still these four forces But there is a special understanding of the interaction force Very integrated (弯曲的这种集合，因为没有人理解，所以他以为有一种引力 。这等同于他的广义相对论。爱因斯坦在 1915 年和 1916 年所做的，就是万有引力几何化了。我认为这是人类历史上的一项伟大发明。因此，之后仍然存在这四种力，但对万有引力有了特殊的理解，非常几何化。)
Then he said again By 1917 and 1918, Einstein said He was not clear at that time Weak force and strong force So he said he wanted to put the electromagnetic force Integrated with external gravitational force Called Unified Field Theory He thought these two forces In fact, the basic structure is the same So from that time to now This Unified Field Theory This goal Became a main goal of basic physics Any achievements? (然后他又说，到了 1917 年和 1918 年，爱因斯坦表示他当时还不清楚弱力和强力。于是他说想把电磁力和万有引力统一起来，称之为统一场论。他认为这两种力实际上基本结构是相同的。所以从那时到现在，这个统一场论的目标成为了基础物理学的一个主要目标。有什么成果吗？)
There are great achievements Now, for the three forces Except the gravitational force, The other three forces, Now more or less Can't say it's completely understood, But more or less It can be said to be united But the last force The gravitational force Still can't be united(现在有了很大的成就。对于三种力，除了万有引力之外，其他三种力或多或少可以说是统一的，尽管不能说完全理解。但最后一种力，万有引力，仍然无法统一。)
So this is also today The first year of the 21st century The most basic problem in theoretical physics Is to ask How to put the gravitational force Also connected with the rest of the force Become a Einstein wanted to do Unified field theory This problem Now there are a lot of articles But more than that If you ask me I think now there is The most important basic concept Has not been invented yet （因此，这也是 21 世纪初年理论物理学中最基本的问题，即如何将万有引力与其他力联系起来，形成一个爱因斯坦曾试图建立的统一场论。这个问题现在已有大量论文探讨，但更重要的是，如果你问我，我认为目前最关键的基础概念尚未被发明出来。）
This is So it can also be said To wait for your generation Or your students Maybe you can have Solve this last problem Of course there are a lot of stories here I don't think it's What ordinary physics can say But The classmate asked just now Say there are these four forces Is not bad Say it needs now Everyone wants to put these four forces It's right to unite But this big job has not been completed yet(因此，可以说这需要等待你们这一代人，或者你们的学生，来解决这个最后的问题。当然，这其中有很多故事，我认为这不是普通物理学能解释的。不过，刚才有同学提到这四种力，这没错。大家都希望能够统一这四种力，但这个重大任务尚未完成。)
But I said last time Because of this kind of thing Many people wrote a lot of books In these books I don't agree Classmates in freshman year Or high school classmates Watch too much I think after reading Not much good And just make everyone feel like Very excited And not solid enough So for example Hawking Hawking was a very famous Astronomical physicist (但我上次说过 因为这种事 很多人写了很多书 这些书我都不认同 大一同学 或者高中同学 看太多 我觉得读完后 没什么好处 反而会让大家都觉得 很兴奋 但不够扎实 比如霍金 霍金是一位非常著名的 天体物理学家)
He has a very important contribution But he wrote a book more than ten years ago It seems to be called History of Time This book I You can borrow it at the library You can turn it over Take it home Can't read for three days Because of this Most of what he said Is some He and Some people's fantasies These fantasies With Mixed together So if you read it There are good and bad Maybe in the future Can't tell which is right Which is right (他做出了非常重要的贡献，但他在十多年前写了一本书，似乎叫《时间简史》。这本书你可以在图书馆借到，可以翻一翻，带回家。三天内可能读不完，因为书中大部分内容是他和一些人的幻想，这些幻想混杂在一起。所以如果你读了，会发现有好有坏，也许将来也无法分辨哪些是对的，哪些是错的。)
This is wrong So I don't agree At your level Look at this kind of book So as for last time I talked about a Green Written Green, he is I want to study String theory Mysterious This mysterious thing Is very mysterious This I think Also at your stage Not worth it Go here OK Then we Please remember Next Monday I have to teach the second time Monday Everyone's CT (这是错误的，所以我不同意。以你的水平看这种书，所以上次我提到的格林写的《格林》，他是我想研究 弦理论 的。这个东西非常玄乎。我认为在你这个阶段也不值得。到这里就好。那么，我们请记住下周一必须交第二次的习题。
弦理论（包括其发展形式超弦理论）的核心目标，正是试图用一个统一的框架来描述和解释自然界的四种基本相互作用力（即引力、电磁力、强力和弱力），从而将描述宏观世界的广义相对论和描述微观世界的量子力学和谐地融为一体。但杨老对弦理论持保留态度，并谨慎看待年轻人投身该领域，主要是基于他对物理学本质的深刻理解以及科学研究的价值判断。他认为物理学的根基在于实验验证，而弦理论在相当长的时间内都缺乏实验支持，且数学结构过于复杂，与已知的物理现实距离较远。